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\usepackage{pandora,euler,amsmath,fullpage}

\begin{document}

\title{Math 2B, Section A, Winter 1997}
\author{Marcel Oliver}
\date{Midterm Exam II}

\maketitle

\noindent Useful integrals:
\begin{gather*}
  \int \frac{du}{\sqrt{1 - u^2}} = \arcsin u + C \\
  \int \frac{du}{1 + u^2} = \arctan u + C \\
  \int \frac{du}{u \sqrt{u^2 - 1}} = \operatorname{arcsec} |u| + C
\end{gather*}
\bigskip

\noindent Some trigonometric identities:
\begin{align*}
  {} &  \sin^2 \theta + \cos^2 \theta = 1
     && \sin 2x = 2 \, \sin x \, \cos x \\
     &  1 + \tan^2 \theta = \sec^2 \theta
     && \cos 2x = 2 \, \cos^2 x - 1 = 1 - 2 \, \sin^2 x \\
     &  1 + \cot^2 \theta = \csc^2 \theta
     && \tan 2x = \frac{2 \, \tan x}{1 - \tan^2 x} \\
     &  \sin^2 x = \frac{1 - \cos 2 x}{2}
     && \cos^2 x = \frac{1 + \cos 2 x}{2} 
\end{align*}
\bigskip

\begin{enumerate}

\item Compute the following integrals.  Be sure to show all your work.
\begin{enumerate}
\item $\displaystyle \int \frac{4x-2}{x^3-x} \, dx$
\item $\displaystyle \int e^x \, \arcsin (e^x) \, dx$
\item $\displaystyle \int \tan^3 z \, dz$
\end{enumerate}
Note: In part (a) use the properties of the logarithm to simplify your
answer.  For parts (b) and (c) some of the identities on the front
page might be useful.
\mbox{ } \hfill (10+10+10)

\item
\begin{enumerate}
\item Find the volume of the solid generated by revolving the graph of
the equation $y = \sin x$ from $x=0$ to $x = \pi$ around the $x$-axis.

\item Compute the arc length of the graph of the $\cosh$ function from
$x=0$ to $x=1$.

The following information might be useful:
\begin{gather*}
  \sinh x  = \tfrac12 \bigl(e^x - e^{-x} \bigr), \quad
  \cosh x  = \tfrac12 \bigl(e^x + e^{-x} \bigr), \\
  \frac{d}{dx} \sinh x = \cosh x, \quad
  \frac{d}{dx} \cosh x = \sinh x, \\
  \cosh^2 x - \sinh^2 x = 1.
\end{gather*}
\end{enumerate}
\hfill (10+10)

\item Consider the function
\[ f(x) = x \, e^{-x}. \]
Sketch the graph of $f$.  Your drawing must show clearly where the
graph is increasing, where decreasing, where concave upward, and where
concave downward.  Locate, identify, and label all extrema and
inflection points.  Find and label all asymptotes.  \hfill (25)


\item Radioactive substances decay according to the equation
\begin{equation*}
  \frac{dN}{dt} = - \lambda \, N, \qquad N(0) = N_0,
\end{equation*}
where the \emph{activity} $N$ describes the number of decays per
second.

\begin{enumerate}
\item Solve the above initial value problem to find an expression for
the activity $N$ in terms of time $t$.

\item Find an expression for the half life of a radioactive substance
in terms of its decay constant $\lambda$.

\item If the initial activity is $N_0 = 1000$ decays per second, and
the half life is $30$s, what is the activity after $90$s?
\hfill (10+10+5)

\item After an accident in a nuclear power plant the activity in the
contaminated area is 100 times the save level.  A week later, the
activity has dropped to 10 times the save level.  Citing the
radioactive decay law, a company spokesman claims that after yet
another week the area will be safe again.  Do you believe this?
Explain your answer.  (This is not a Math question---extra credit!)
\end{enumerate}

  
\end{enumerate}

\end{document}
